Robust and efficient statistical learning algorithms with applications in actuarial science

大数据时代代表着统计方法通过与包括精算科学在内的广泛领域相关的应用而大放异彩的机会。然而，为了抓住并最大限度地利用这一机遇，研究人员和从业者必须有效地管理大数据带来的挑战。首先，数量当然不意味着质量，数据也不例外。在有严重错误的劣质数据中识别趋势是困难的。对异常值具有健壮性的方法有助于完成这项任务。我的研究目标之一是引入对精算师具有实用意义的稳健贝叶斯模型。例如，广义线性模型(GLM)在一般保险索赔建模中普遍存在。鲁棒广义似然模型是一类拟提出的模型。当离群值移动得越来越远时，它们将具有一个显著的特征，即完全基于极限中的非离群值来产生结果，而在没有离群值的情况下执行与传统模型相似的性能。事实上，异常值的影响逐渐消失，反映出在它们距离大部分数据不远的一开始，它们是否真的是异常值存在不确定性。自动处理这种不确定性的方法在高维和变量选择问题中特别有价值。其次，提出处理严重误差的复杂模型是不够的。推理所需的数值方法必须根据模型的复杂性和数据大小进行调整，以确保统计程序是可执行的。为此，我将提出自动不可逆跳转算法。这些方法是马尔可夫链蒙特卡罗(MCMC)方法，用于逼近关于模型及其参数的联合后验分布的积分，允许同时选择贝叶斯变量和参数估计。与可逆方法相比，不可逆方法在典型情况下以其更好的可扩展性而闻名。这是由于具有以持续移动为特征的路径的马尔可夫链，其允许更快地遍历状态空间，并防止通常由可逆方案表现出的扩散行为。这转化为较少的自相关，这意味着从后验分布获得独立样本的迭代次数较少。因此，结果更接近于常规的蒙特卡罗，这是MCMC的最终目标。稳健的统计模型与自动和高效的自动推理数值方法相结合，产生了稳健和高效的统计学习算法。统计程序(例如，相对于典型的机器学习算法)的优势在于，它们允许将风险和不确定性量化。这种量化是精算师作用的核心，并允许发布包含关于保险公司支付未来索赔能力的丰富概率描述的报表。

The big data era represents an opportunity for statistical methods to shine, through applications relevant to a wide spectrum of fields, including actuarial science. In order to seize and make the most out of this opportunity, researchers and practitioners must, however, effectively manage the challenges that big data pose. Firstly, quantity surely does not imply quality, and data are no exception to this rule. Identifying trends in poor quality data with gross errors is difficult. Methods that are robust against outliers help in this task. One of my research goals is to introduce robust Bayesian models of practical relevance for actuaries. Generalised linear models (GLMs) are, for instance, ubiquitous in general insurance claim modelling. Robust GLMs are a class of models to be proposed. They will have the remarkable characteristic of producing results based solely on the nonoutliers in the limit when the outliers move further and further away, while performing similarly to the traditional models in the absence of outliers. The impact of the outliers in fact gradually vanishes, reflecting that at the beginning when they are not so far from the bulk of the data, there is an uncertainty about whether they really are outliers or not. Methods automatically dealing with this uncertainty are particularly valuable in high-dimensional and variable selection problems. Secondly, proposing complex models handling gross errors is not enough. The numerical methods required for inference must scale with the model complexity and data size to ensure that the statistical procedures are implementable. For this purpose, I will propose automatic nonreversible jump algorithms. These are Markov chain Monte Carlo (MCMC) methods used for approximating integrals with respect to joint posterior distributions of models and their parameters, which allows simultaneous Bayesian variable selection and parameter estimation. Nonreversible methods are known for their better scalability in typical cases, comparatively to their reversible counterparts. This is due to Markov chains with paths characterised by persistent movement that allow to traverse the state space more quickly and prevent the diffusive behaviour often exhibited by reversible schemes. This translates into less autocorrelations, which implies less iterations to obtain independent samples from the posterior distributions. The results are thus closer to regular Monte Carlo, which is the ultimate MCMC goal. Robust statistical models together with automatic and efficient numerical methods for automated inference result in: robust and efficient statistical learning algorithms. The advantage of statistical procedures (over typical machine learning algorithms for instance) is that they allow risk and uncertainty quantification. This quantification is at the core of actuaries' role and allows to issue statements containing rich probabilistic descriptions about the capacity of insurance firms to pay for future claims.